Downing Street said the Barnett formula does not apply to the new money as it is being provided as an addition to the Northern Ireland Executive’s block grant.

In a similar way, the formula did not apply to city deals in Scotland and Wales, or previous packages of support for Northern Ireland, a Number 10 source said.

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David Mundell has previously said he would oppose any deal that “deliberately sought to subvert the Barnett rules”

The SNP said the “grubby” deal between the DUP and the government would see Scotland being thrown “little more than scraps from the table”, and demonstrated “how little the Tories care about Scotland”.

The party’s Westminster leader, Ian Blackford, added: “Only 24 hours ago, David Mundell was categorically assuring us that Scotland would be in line for Barnett consequentials as a result of the DUP deal.

“So he has seemingly either been deliberately misleading people, or he is completely out of the loop even in Theresa May’s crumbling government.

“This was the first big test of the new Scottish Tory MPs, but despite all of their bluster, they clearly have no authority and no influence – and now they have no credibility.”

‘Shameless deal’

Scottish Labour leader Kezia Dugdale warned that the prime minister risked “weakening the bonds that unite the UK” if she did not provide extra money to tackle austerity in Scotland, Wales and the regions of England.

Carwyn Jones, the Labour first minister of Wales, said the agreement “further weakens the UK” and “all but kills the idea of fair funding for the nations and regions.”

And Scottish Greens co-convener Patrick Harvie called on Mr Mundell to resign as secretary of state for Scotland over the government’s “shameless deal with the homophobes and climate deniers of the DUP”.

What is the Barnett formula?

Since the late 1970s the Barnett formula has been used to determine annual changes in the block grant to each nation of the UK

When there is a change in funding for devolved services in England, for example health or education, the Barnett formula aims to give each country the same pounds-per-person change in funding

But the formula is not set out in law, and in practice the Treasury decides how to apply it

The UK government also provides other grants to devolved administrations outside of the block grant, which are not covered by Barnett

These grants are for less predictable demand driven spending, and are negotiated between the UK government and devolved administrations

The deal, which comes two weeks after the election resulted in a hung parliament, will see the 10 DUP MPs back the Tories in key Commons votes.

Theresa May fell nine seats short of an overall majority after the snap election, meaning she is reliant on other parties to pass legislation, including relating to the terms of the UK’s withdrawal from the EU.

DUP leader Arlene Foster insisted the “wide-ranging” agreement was “good for Northern Ireland and for the UK” and predicted it would deliver a stable government as the country embarks on the Brexit process.

Mrs Foster said the money was needed to address the challenges from Northern Ireland’s “unique history”.

The prime minister said the pact was a “very good one” for the UK as a whole, adding: “We share many values in terms of wanting to see prosperity across the UK, the value of the union, the important bond between the different parts of the UK”.

Where will the £1bn of extra money to Northern Ireland go?

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Health: A minimum of £250m, with £200m directed to health service transformation and £50m towards mental health provision. It will also receive £50m to “address immediate pressures”

Education: £50m to “address immediate pressures”

Infrastructure: £400m for projects including delivery the York Street Interchange, plus £150m to provide ultra-fast broadband across Northern Ireland.

Deprivation: £100m over five years targeted to deprived communities

VAT and Air Passenger Duty tax: Agreed to further consultation

Corporation tax: Agreed to work towards devolving the tax to Stormont

City deals and Enterprise Zones: Agreed to “comprehensive and ambitious set” of city deals and “limited number” of Enterprise Zones